Lp operator algebras
with approximate identities and real positivity I

David P. Blecher
Department of Mathematics, University of Houston,
Houston, TX 77204-3008
[
and N. Christopher Phillips
Department of Mathematics, University of Oregon,
Eugene, OR 97403-1222
[

Abstract.

We initiate an investigation into how much the existing theory
of (nonselfadjoint) operator algebras on a Hilbert space
generalizes to algebras acting on Lp spaces.
In particular we investigate the applicability
of the theory of real positivity,
which has recently been useful in the study of
L2-operator algebras and Banach algebras,
to algebras of bounded operators on Lp spaces.

This work is based on work supported by
the US National Science Foundation under
Grant DMS-1501144 (Phillips), a Simons Foundation grant 527078 (Blecher),
and a minigrant from the Mathematics
Department of the University of Houston. Material from this paper
and its sequel were presented at 2017 conferences in Houston
(August), the East Coast Operator Algebras Symposium, and the SAMS congress.

1. Introduction

In a series of recent papers
(see e.g. [41, 42, 43, 44])
the second author has
pointed out that the study
of algebras of bounded operators on Lp spaces,
henceforth,
Lp-operator algebras, has been somewhat overlooked, and
has initiated the study of these objects.
Subsequently others have followed him into this inquiry
(for example,
Gardella, Thiel, Lupini, and Viola;
see e.g. [22, 23, 25, 21, 46]).
However, as he has frequently stated,
these investigations have been very largely focused on examples;
one still lacks an abstract general theory in this setting.

Here and in a sequel in preparation we initiate an investigation into
how much the existing theory of
(nonselfadjoint) L2-operator algebras
(see e.g. [6], [8])
generalizes to the Lp case.
We restrict ourselves almost
exclusively to the
“isometric theory”; perhaps we may pursue the isomorphic theory
elesewhere.
In addition to establishing some general facts
about Lp operator algebras,
the main goal of the present paper
is to investigate to what extent the first author’s theory
of real positivity
(developed with Read, Neal, Ozawa, and others;
see e.g. [8, 9, 10, 7]),
is applicable to
Lp-operator algebras,
particularly those with contractive approximate identities.
As an easy motivation,
notice that the canonical approximate identity
for K(lp) is real positive,
and the real positive elements span B(Lp([0,1]))
(as they do any unital Banach algebra).
The theory of real positivity was developed as a tool for
generalizing certain parts of C∗-algebra theory
to more general algebras.
In [7] (see also [5])
this was extended to Banach algebras,
and of course therefore applies to Lp-operator algebras.
However, some parts of [7]
applied only to certain classes of Banach algebras defined there,
which were shown to behave in some respects
similarly to L2-operator algebras.
For example,
a nonunital approximately unital Banach algebra A
was defined there to be
scaled if the set of restrictions to A
of states on A1 equals the quasistate space Q(A) of A
(that is, the set of λφ for λ∈[0,1]
and φ a norm 1 functional on A
that extends to a state on A1).
All unital Banach algebras are scaled.
In [7],
there are several pretty equivalent conditions
for a Banach algebra to be scaled,
and this class of Banach algebras
was shown to have several nice theoretical features,
such as a Kaplansky density type theorem.
Thus it is natural to ask the following:

To which of the classes
defined in [7] do Lp operator algebras belong?

For those classes in [7] to which they do not belong,
to what extent do the theorems for those classes from [7]
still hold for Lp-operator algebras?

To what extent do other parts of the theory of L2-operator algebras
hold for Lp-operator algebras?

We focus mostly here on
the parts of the theory of the first author
with Read, Neal, and others referred to above
that were not already extended to the general classes
considered in [7].
For example, one may ask if the material
in Section 4 in [7],
and in particular the theory of hereditary subalgebras, improves
(that is, becomes closer to the L2-operator case) for
Lp-operator algebras.
Similarly, one may ask about the
noncommutative topology
(in the sense of Akemann, Pedersen, L. G. Brown, and others) of
Lp-operator algebras.
In papers of the first author
with Read, Neal, and others referred to above, Akemann’s noncommutative
topology of C∗-algebras
was fused with the classical theory of (generalized) peak sets of
function algebras to create
a relative noncommutative topology for closed subalgebras of
C∗-algebras that has proved
to have many applications.
Examples given in [7] show that
not much of this will extend to general Banach algebras, and it is
natural to ask if Lp-operator algebras
are better in this regard.
Most of the present paper and the sequel in preparation is devoted to
answering these questions.
In the process we answer some open questions from [7].

We admit from the outset that for p≠2,
and for some significant part of the theory,
the answer to question (2) above
is so far in the negative.
This may
change somewhat in the future,
for example if we were able to prove the above mentioned variant of
Kaplansky’s density theorem for
all approximately unital Lp-operator algebras.
It should also be
admitted that for p≠2
the “projection lattice” of
B(Lp(X,μ)) is problematic from the perspective of our paper (see
Example 3.2 and the sequel paper),
in contrast to the projection lattice of von Neumann algebras and
L2-operator algebras.
Concerning question (1), the classes of scaled
and M-approximately unital Banach algebras
defined in [7]
coincide for Lp-operator algebras
with a contractive approximate identity.
However some Lp-operator algebras
with contractive approximate identities
are scaled
and others are not.
This answers the question from [7]
as to whether every Banach algebra
with a contractive approximate identity is scaled.
Also, non-scaled Lp-operator algebras
with a contractive approximate identity
may contain no real positive elements
(whereas it was shown in [7]
that if they are scaled
then there is an abundance of real positive elements,
e.g. every element in A is a difference of two
real positive elements).

Concerning question (3) above,
indeed some aspects of the theory improve.
For example,
Section 4 of [7] improves drastically in our setting, and indeed
Lp-operator algebras do support a basic theory of
noncommutative topology and hereditary subalgebras, unlike general
Banach algebras.
This is worked out in the sequel paper in preparation, where the reader
will find many more positive results than in the present paper.
It is worth remarking that the methods used here do not seem to extend
far beyond the
class of Lp-operator algebras as we will discuss elsewhere.
However, most of our results for Lp-operator algebras
in Sections 2 and 4 do generalize to the class
of SQp-operator algebras, that is closed
algebras of operators on an SQp space, that is,
a quotient of a subspace of an Lp space.
(See e.g. [33].
We thank Eusebio Gardella for suggesting SQp spaces
after we
listed in a talk the properties needed for our
results to work.)

On the other hand,
except cosmetically, not much to speak of
in Section 3 of [7] improves for Lp-operator algebras,
in the sense of becoming significantly more
like the L2-operator algebra case.
However several of the concepts appearing throughout [7]
become much simpler in our setting.
For example as we said above,
two of the main classes of Banach algebras considered there coincide.
Also as we shall see the subscript and superscript e
which appear often in [7] may be erased in our setting,
since we are able to show that all Lp-operator algebras
are Hahn-Banach smooth.
Then of course the Arens regularity of Lp-operator algebras
means that many irritating features of the bidual
appearing in [7]
disappear, such as mixed identities in A∗∗.

We now describe the contents of our paper.

We will be assuming that p∈(1,∞)∖{2} in all
results in the paper unless
stated to the contrary.
As usual 1p+1q=1.
In the remainder of Section 1
we give some notation and basic definitions.
In Section 2 we discuss some notation and background,
together with some useful general facts, concerning topics
such as dual and bidual algebras, the multiplier unitization, states
and real positivity, representations, etc.
Many of the facts in this section are known.
In Section 3
we list the main examples of Lp-operator algebras
which we use in this paper for counterexamples,
as well as some other basic examples not in the literature.
Section 4 contains miscellaneous results on quotients,
Meyer’s unitization theorem,
the Cayley and F transform,
support idempotents of elements,
and some important consequences for
us of the strict convexity of Lp spaces,
in particular that Lp-operator algebras are Hahn-Banach smooth.
In Section 5 and Section 6
we discuss M-ideals and scaled Banach algebras.
In the sequel paper in preparation
we show that the theory of one-sided ideals,
hereditary subalgebras, open projections, etc. for Lp-operator algebras
is quite similar to the (nonselfadjoint) L2-operator algebra case.
This is particularly so for certain large
classes of Lp-operator algebras.
We feel that this is important, since hereditary subalgebras play
a large role in modern C∗-algebra theory, and thus hopefully will be
important for Lp-operator algebras too.

We end our introduction with a few definitions and basic lemmas.

We set R+=[0,∞).

Notation 1.1.

Let E be a normed vector space.
Then Ball(E) is the closed unit ball of E,
that is,

Ball(E)={ξ∈E:∥ξ∥≤1}.

Notation 1.2.

Let p∈[1,∞].
Let E and F be normed vector spaces.
We denote by E⊕pF their Lp direct sum,
that is,
the algebraic direct sum E⊕F
with the norm given for ξ∈E and η∈F
by
∥(ξ,η)∥=(∥ξ∥p+∥η∥p)1/p
if p<∞
and ∥(ξ,η)∥=max(∥ξ∥,∥η∥)
if p=∞.

Although many of our Banach algebras
have identities of norm greater than 1,
the adjectives “unital” or “approximately unital” for a Banach
algebra will carry a norm 1 requirement.

Definition 1.3.

A unital Banach algebra
is a Banach algebra with an identity 1
such that ∥1∥=1.

Definition 1.4.

A cai in a Banach algebra
is a contractive approximate identity,
that is, an approximate identity (et)t∈Λ
such that ∥et∥≤1
for all t∈Λ.
An approximately unital Banach algebra
is a Banach algebra which has a cai.

When we write Lp or Lp(X)
we mean the Lp space of some measure space (X,μ).

Recall that a Banach space E is strictly convex
if whenever ξ,η∈E∖{0}
satisfy
∥ξ+η∥=∥ξ∥+∥η∥,
then there is λ∈(0,∞)
such that ξ=λη,
and smooth
if for given ξ∈E with ∥ξ∥=1,
there is a unique η∈Ball(E∗)
with ⟨ξ,η⟩=1.
If 1<p<∞,
then Lp(X) is strictly convex
(by the converse to Minkowski’s inequality).
Moreover,
still assuming 1<p<∞,
the space Lp(X) is smooth,
with η above given by the function

η(x)={¯¯¯¯¯¯¯¯¯¯ξ(x)|ξ(x)|p−2ξ(x)≠00ξ(x)=0

in Lq(X).
We will frequently use the fact that Lp(X) is smooth
and strictly convex if 1<p<∞.

Definition 1.5.

Let p∈[1,∞).
An Lp-operator algebra
is a Banach algebra which is isometrically isomorphic
to a norm closed subalgebra of the algebra of bounded operators
on Lp(X,μ)
for some measure space (X,μ).
When p=2 we simply refer to an operator algebra.
(See the beginning of Section 2.1 of [6],
except that we do not consider matrix norms
in the present paper.)

Definition 1.6.

Let A be an Lp-operator algebra
(not necessarily approximately unital).
We say that an Lp operator algebra B
is an Lp operator unitization of A
if either A is unital and B=A,
or if A is nonunital,
B is unital (in particular, by our convention, ∥1∥=1),
and A is a codimension one ideal in B.

Let A be a nonunital approximately unital Banach algebra
(as in Definition 1.4).
We define its multiplier unitizationA1
to be the usual unitization A+C⋅1 with the norm

∥a+λ1∥A1=sup({∥ac+λc∥:c∈Ball(A)}).

for a∈A and λ∈C.
If A is already unital then we
set A1=A.

Remark 1.8.

We recall the following easy standard facts.

If A is an approximately unital Banach algebra,
then the standard inclusion of A in A1 is isometric.

Let A be a Banach algebra,
and let (et)t∈Λ be any cai in A.
Then

∥a+λ1∥A1=limt∥aet+λet∥=supt∥aet+λet∥.

If A is any nonunital Banach algebra,
and B is a unital Banach algebra which contains A as
a codimension 1 subalgebra,
then the map χ0:B→C,
given by
χ0(a+λ1B)=λ
for a∈A and λ∈C,
is contractive.

If A is any nonunital Banach algebra with a cai,
and B is a unital Banach algebra which contains A as
a codimension 1 subalgebra,
then the map ψ:B→A1,
given by
ψ(a+λ1B)=a+λ1A1
for a∈A and λ∈C,
is a contractive homomorphism.
Thus A1 has the smallest norm of
any unitization.
This follows e.g. by a small variant of the proof
of Lemma 1.9 below.

Lemma 1.9.

Suppose that A is a closed subalgebra
of a nonunital approximately unital
Banach algebra B,
and suppose that A has a cai but is not unital.
Then for all a∈A and λ∈C
we have
∥a+λ1∥A1≤∥a+λ1∥B1.

Proof.

Clearly

sup({∥ac+λc∥:c∈Ball(A)})≤sup({∥ac+λc∥:c∈Ball(B)}),

as desired.
∎

It is easy to find examples showing that the homomorphism above need
not be isometric, for example, with notation as in
Example 3.2 (or Example 3.5) below,
Ce2⊗c0⊆Mp2⊗c0.
However we have the following result.

Lemma 1.10.

Let A and B be nonunital
approximately unital Banach algebras.
Let φ:A→B be a contractive
(resp. isometric) homomorphism.
Suppose that there is a cai (et)t∈Λ for A
such that
(φ(et))t∈Λ is a cai for B.
Then the obvious unital homomorphism A1→B1
between the multiplier unitizations is contractive (resp. isometric).

Proof.

If a∈A and λ∈C
then

∥π(a)π(et)+λπ(et)∥≤∥aet+λet∥.

In the isometric case this is an equality.
Taking limits over t and using
Remark 1.8 (2)
gives the result.
∎

We recall two further standard facts.
The first is that the relation
K(L2(X))∗∗=B(L2(X)) is true
with 2 replaced by any p∈(1,∞).

Theorem 1.11.

There is an isometric isomorphism
K(Lp(X,μ))∗→Lq(X,μ)ˆ⊗Lp(X,μ)
which for ρ∈Lp(X,μ)
and η∈Lq(X,μ)
sends η⊗ρ to the operator
ξ↦⟨ξ,η⟩ρ.

There is an isometric algebra isomorphism
from K(Lp(X,μ))∗∗ (with either Arens multiplication)
to B(Lp(X,μ)) which extends the
inclusion K(Lp(X,μ))⊆B(Lp(X,μ)).

Proof.

This follows from results of Grothendieck,
as described in the theorem on page 828 of [38]
and the discussion afterwards.
See also the discussion on page 24, Corollary 4.13,
and Theorem 5.33 of [49] (and we thank M. Mazowita for
this reference). The explicit
formulae there make it easy to check the Arens
product assertion.
One needs to know that Lp(X,μ) has the Radon-Nikodym property,
and this follows e.g. from [49, Corollary 5.45].
∎

By Theorem 1.11,
a net (xt)t∈Λ in B(Lp(X))
converges weak* to x if and only if,
with 1p+1q=1,

∞∑k=1⟨xtξk,ηk⟩→∞∑k=1⟨xξk,ηk⟩

for all ξ1,ξ2,…∈Lp(X)
and η1,η2,…∈Lq(X)
with ∑∞k=1∥ξk∥p∥ηk∥q<∞
(or equivalently, by the usual trick,
with ∑∞k=1∥ξk∥pp<∞
and ∑∞k=1∥ηk∥qq<∞).
If (xt)t∈Λ is bounded then by Banach duality principles
this is equivalent to xt→x in the weak operator topology,
that is
⟨xtξ,η⟩→⟨xξ,η⟩
for all ξ∈Lp(X) and η∈Lq(X).
We will not use this here
but it is well known that essentially the usual L2 operator proof
shows that the weak operator closure
of a convex set in B(Lp([0,1]))
equals the strong operator closure.
Indeed, for a Banach space E,
the strong operator continuous linear functionals on B(E)
are the same as those that are weak operator continuous.

The argument for the following well known lemma
will be reused
several times,
once in the form of an approximate identity bounded by M
converging weak* to an identity in A∗∗ of norm at most M.

Lemma 1.12.

Let A be an approximately unital Arens regular Banach algebra.
Then A∗∗ has an identity 1A∗∗ of norm 1,
and any cai for A converges weak* to 1A∗∗.

Proof.

The argument follows the proof of [6, Proposition 2.5.8].
Since identities are unique if they exist,
it suffices to show that every subnet of any cai in A
has in turn a subnet which converges to an identity for A∗∗.
Using Alaoglu’s Theorem and since a subnet of a cai is a cai,
one sees that it is enough to show that if
e∈A∗∗ is the weak* limit of a cai,
then e is an identity for A∗∗.
Multiplication on A∗∗ is separately weak* continuous
by [6, 2.5.3],
so ea=ae=a for all a∈A.
A second application of separate weak* continuity of multiplication
shows that this is true for all a∈A∗∗.
∎

Proof.

We first recall (Theorem 3.3 (ii) of [29], or [33],
or the
remarks above Theorem 4.1 in [18])
that any ultrapower of Lp spaces (resp. SQp spaces)
is again an Lp space (resp. SQp space).
In the SQp space case this uses the well
known fact that ultrapowers behave well
with respect to subspaces and quotients (see e.g. [29]).
In particular,
such an ultrapower is reflexive,
so
every Lp space (resp. SQp space) is superreflexive.
(See Proposition 1 of [17].)

Now let E be an Lp space (resp. SQp space).
Theorem 1 of [17]
implies that B(E)
is Arens regular.
The proof of Theorem 1 of [17]
embeds B(E)∗∗ isometrically as a subalgebra
of B(F)
for a Banach space F
obtained as an ultrapower of lr(E)
for an arbitrarily chosen r∈(1,∞)
(called p in [17]).
We may choose r=p.
Then lr(E) is isometrically isomorphic
to an Lp space (resp. SQp space).
Since ultrapowers of Lp spaces
(resp. SQp spaces)
are Lp spaces (resp. SQp spaces)
as we said at the start of this proof,
we have shown that B(E)∗∗
is an Lp- (resp. SQp-) operator algebra.

Now suppose that A⊆B(E)
is a norm closed subalgebra.
Since B(E) is Arens regular,
A∗∗ is a subalgebra of B(E)∗∗
and A is Arens regular
by 2.5.2 in [6].
It is now immediate that A∗∗ is an Lp-
(resp. SQp-) operator algebra.
It also follows from 2.5.3 in [6]
that multiplication on A∗∗ is separately weak* continuous.
∎

It follows from [17, Proposition 8]
that B(L1(X,μ)) is not Arens regular
unless L1(X,μ) is finite dimensional.

Corollary 2.2.

Let p∈(1,∞)
and let (X,μ) be a measure space.
Then multiplication on B(Lp(X,μ))
is separately weak* continuous.

Proof.

Definition 2.3.

Let p∈(1,∞).
A dual Lp-operator algebra
is a Banach algebra A with a predual
such that there is a measure space (X,μ)
and an isometric and weak* homeomorphic isomorphism
from A to a weak* closed subalgebra of B(Lp(X,μ)).

By Corollary 2.2, the multiplication
on a dual Lp-operator algebra is separately weak* continuous.

Corollary 2.4.

Let p∈(1,∞)
and let A be an Lp-operator algebra.
Then A∗∗ is a dual Lp-operator algebra.

Proof.

The embedding of B(Lp(X,μ))∗∗
in Lemma 2.1 coming from the proof from [17]
is easily checked
to be weak* continuous,
hence a weak* homeomorphism onto its range by the Krein-Smulian theorem.
Hence B(Lp(X,μ))∗∗ is a dual Lp-operator algebra.
It easily follows
that A∗∗ is too.
∎

Lemma 2.5.

Let p∈(1,∞)
and let A be a dual Lp-operator algebra.
Then:

The weak* closure of any subalgebra of A
is a dual Lp-operator algebra.

If A is approximately unital
then A is unital.

Proof.

The proofs are essentially the same as in the case p=2,
as done in the proof of Proposition 2.7.4 in [6].
∎

2.2. States, hermitian elements,
and real positivity

Definition 2.6.

If A is a unital Banach algebra,
then a state on A
is a linear functional ω:A→C
such that ∥ω∥=ω(1)=1.
If A is an approximately unital Banach algebra,
we define a state on A
to be a linear functional ω:A→C
such that ∥ω∥=1
and ω is the restriction to A of a state on the
multiplier unitization A1
(Definition 1.7).

We denote by S(A) the set of all states on A, and write
Q(A) for the quasistate space
(that is, the set of λφ for λ∈[0,1]
and φ∈S(A)).

If e=(et)t∈Λ is
a cai for A, define

Se(A)={ω∈Ball(A∗):ω(et)→1}

and define

Qe(A)={λφ:λ∈[0,1] and φ∈Se(A)}.

If A is a C∗-algebra (unital or not),
this definition gives the usual states and quasistates on A.

The first part of the following definition
is Definition 2.6.1 of [39].

Definition 2.7.

Let A be a unital Banach algebra,
and let a∈A.
We define the numerical range of a to be
{φ(a):φ∈S(A)}.

If E is a Banach space and a∈B(E),
we define the spatial numerical range of a to be

{⟨aξ,η⟩:ξ∈Ball(E) %
and η∈Ball(E∗) with ⟨ξ,η⟩=1}.

There are other definitions of the numerical range.
For our purposes, only the convex hull is important,
and by Theorem 14 of [36]
the convex hulls of the
numerical range and the spatial numerical range of
an element in B(E) are always the same.

Let A be a unital Banach algebra,
and let a∈A.
Then a is hermitian if and only if
φ(a)∈R for all states φ of A.

Lemma 2.10.

Let A be an approximately unital Banach algebra,
and let B⊆A be a closed subalgebra
which contains a cai for A.
Let a∈B.
Then a is hermitian as an element of B
if and only if a is hermitian as an element of A.

Proof.

By definition,
we work in the multiplier unitizations.
By Lemma 1.10,
B1 is isometrically a unital subalgebra of A1.
The Hahn-Banach Theorem now shows that states on B1
are exactly the restrictions of states on A1.
So the conclusion follows from Lemma 2.9.
∎

Let (X,μ) be a measure space that is not σ-finite.
Recall that a function f:X→C
is locally measurable if f−1(E)∩F is measurable
for all Borel sets E⊆C and
all subsets F⊆X of finite measure.
Two such functions are “locally a.e. equal”
if they agree a.e. on any such set F.
We interpret L∞(X,μ) as
L∞loc(X,μ),
the Banach space of
essentially bounded locally measurable scalar functions
mod local a.e. equality.

Further recall that a measure space (X,μ) is decomposable
if X may be partitioned into sets Xi
of finite measure for i∈I
such that a set F in X is measurable if and only if F∩Xi is
measurable for every i∈I, and then
μ(F)=∑i∈Iμ(F∩Xi).
By e.g. p. 136 in [32],
any abstract Lp space “is” decomposable, indeed it is isometric
to a direct sum of Lp space of finite measures.
Thus, we may assume that all measure spaces (X,μ) are decomposable.

Proposition 2.11.

Let p∈[1,∞)∖{2}.
Let (X,μ)
be a decomposable measure space,
and let a∈B(Lp(X,μ))
be hermitian.
Then there is a real valued function
f∈L∞(X,μ)
such that a is multiplication by f,
and such that |f(x)|≤∥a∥ for all x∈X.

This is well known under certain conditions such as if
μ is σ-finite and p is not an even number.
(See e.g. Theorem 4 and the
remark following it in [52].)
It is probably folklore
in the general case, but we are not aware of the
latter in the literature.

Let X=∐i∈IXi
be a partition of X into sets of finite measure
as in the discussion of decomposability above.
For i∈I let ei∈B(Lp(X,μ))
be multiplication by χXi.
Since hermitian elements have numerical range contained in R,
we can apply Theorem 6
of [40]
(see the beginning of [40] for the definitions and notation),
to see that a commutes with ei for all i∈I.
One easily checks that h=eiaei is a hermitian element
of B(Lp(Xi,μ)). If p is not an even integer,
then it is known that lp doesn’t
contain a two dimensional Hilbert space, and so
Theorem 4 of [52]
implies that there is real valued function
fi∈L∞(Xi,μ)
such that h is multiplication by fi. The same reference also
does the case that μ has no atomic part in Xi. For the general case
the same result is deduced in [25] from the finite measure case of Lamperti’s theorem
by considering the invertible isometries eith for t∈[0,1].

We can clearly assume that
fi is bounded by ∥eiaei∥≤∥a∥.
Now define f:X→R
by f(x)=fi(x) when i∈I and x∈Xi.
Then f is bounded by ∥a∥,
and is measurable by the choice of the partition of X.
For i∈I and ξ∈Lp(Xi,μ)
we clearly have aξ=fξ.
It follows from density of the linear span
of the subspaces Lp(Xi,μ) that a is multiplication by f.
∎

Definition 2.12.

Let A be a unital Banach algebra.
Let a∈A.
We say that a is accretive
or real positive
if the numerical range of a
is contained in the closed right half plane.
That is,
Re(φ(a))≥0
for all states φ of A.

If instead A is approximately unital,
we define the real positive elements of A
to be the elements in A which are
real positive in the multiplier unitization A1.

In both cases,
we denote the set of real positive elements of A
by rA.

For other equivalent conditions for real positivity,
see for example [5, Lemma 2.4 and Proposition 6.6].

We warn the reader that rA∗∗ is
defined after Lemma 2.5 of [7] to be a proper subset
of the real positive elements in A∗∗,
the set of
elements of A∗∗ which are real positive
with respect to (A1)∗∗.
One should be careful with this ambiguity; fortunately it only
pertains to second duals
and seldom arises.
(Also see Proposition 4.26.)

Lemma 2.13.

Let A be an approximately unital Banach algebra,
and let B⊆A be a closed subalgebra
which contains a cai for A.
Let a∈B.
Then a is real positive as an element of B
if and only if a is real positive as an element of A.

Proof.

The proof is the same as that of Lemma 2.10,
using Definition 2.12
in place of Lemma 2.9.
∎

Lemma 2.14.

Let p∈[1,∞)∖{2},
let A be an approximately unital Lp operator algebra,
and assume that the multiplier unitization A1
is again an Lp operator algebra.
Let a∈A be hermitian.
Then there exist b,c∈A,
each of which is both hermitian and real positive,
such that

(2.1)

a=b−c,bc=cb=0,∥b∥≤∥a∥,and∥c∥≤∥a∥.

By Lemma 2.23 below,
the hypothesis that A1 be an Lp operator algebra
is automatic for p≠1.

It seems unlikely
that this result holds for a general Banach algebra.

We may assume (using e.g. p. 136 in [32])
that (X,μ) is a decomposable measure space
and A1 is a unital subalgebra of B(Lp(X,μ)).
Since a is hermitian in A1,
Lemma 2.10 implies that
a is hermitian in B(Lp(X,μ)).
Proposition 2.11
provides f∈L∞(X,μ)
such that a is multiplication by f,
and such that |f(x)|≤∥a∥ for all x∈X.

Choose a sequence (rn)n∈N
of polynomials with real coefficients
such that rn(λ)→λ1/4
uniformly on [0,∥a∥].
Adjusting by constants and scaling,
we may assume that rn(0)=0
and |rn(λ)|≤∥a∥1/4
for λ∈[0,∥a∥].
Set sn(λ)=rn(λ2)2
for λ∈[−∥a∥,∥a∥].
Then (sn)n∈N
is a sequence of polynomials with real coefficients
such that rn(λ)→λ uniformly on [−∥a∥,∥a∥].
Moreover, for all n∈N
we have sn(0)=0
and 0≤sn(λ)≤∥a∥
for all λ∈[−∥a∥,∥a∥].
In particular,
sn∘f→|f| uniformly on X.

For n∈N,
define dn=sn(a),
which is the multiplication operator by the function sn∘f,
and let d be the multiplication operator by |f|.
Then dn∈A for all n∈N and ∥dn−d∥→0,
so d∈A
and ∥d∥≤∥a∥.
Therefore also

The multiplication operator map
from L∞(X,μ) to B(Lp(X,μ))
is an isometric unital homomorphism.
(Recall the convention that
we are using L∞loc(X,μ) here.)
The functions 12(|f|+f)
and 12(|f|−f)
are nonnegative,
hence both hermitian and real positive in L∞(X,μ)
(because L∞(X,μ) is a C∗-algebra).
Lemma 2.10 and Lemma 2.13
therefore imply that their multiplication operators b and c
are both hermitian and real positive in B(Lp(X,μ)).
A second application of these lemmas
shows that the same holds in A1.
By definition, this is also true in A.
∎

Definition 2.15.

Let A be a unital or approximately unital Banach algebra.
Taking 1 to be the identity of A1
in the approximately unital case,
we define
FA={a∈A:∥1−a∥≤1}.

Let A be a unital or approximately unital Banach algebra.
Then, in the notation of Definition 2.12
and Definition 2.15,
we have rA=¯¯¯¯¯¯¯¯¯¯¯¯¯¯R+FA.

We recall same facts about roots of elements of rA.

Definition 2.17.

Let A be a unital or approximately unital Banach algebra,
let b∈rA, and let t∈(0,1).
If A is unital,
we denote by bt
the element bt constructed in [34, Theorem 1.2].
If A is approximately unital,
let A1 be the multiplier unitization,
recall that b∈rA1 by definition,
and define bt to be as above but evaluated in A1.

The conditions required in [34, Theorem 1.2]
are weaker than here,
but this case is all we need.
Such noninteger powers,
for the special case ∥b−1∥<1 and when A is commutative,
seem to have first appeared in Definition 2.3 of [20].
A discussion relating this definitions to others,
and giving a number of properties,
is contained in [7],
from Proposition 3.3 through Lemma 3.8 there.
In particular, (b1/n)n=b and t↦bt is continuous.
For later use, we recall several of these properties
and state a few other facts not given explicitly in [7].

Proposition 2.18.

Let A be a unital or approximately unital Banach algebra,
and let a∈rA.

If t∈(0,1) and ∥b−1∥≤1
(that is, b∈FA),
then

bt=1+∞∑k=1t(t−1)(t−2)⋯(t−k+1)k!(−1)k(1−b)k,

with absolute convergence.

If t∈(0,1) and λ∈(0,∞)
then (λx)t=λtxt.

For all t∈(0,1),
∥at∥≤2∥a∥t/(1−t).

For all t∈(0,1),
at is a norm limit of polynomials in a with no constant term.

For all t∈(0,1),
ata=aat.

limn→0∥a1/na−a∥=limn→0∥aa1/n−a∥=0.

If a∈FA and t∈(0,1),
then ∥1−at∥≤1.

Proof.

For part (1),
see the proof of [7, Proposition 3.3]
and the discussion in and before
the Remark before [7, Lemma 3.6].

Lemma 2.19.

Suppose that A is a closed subalgebra of an approximately unital
Banach algebra B, and suppose that A has a cai.
Then FB∩A⊆FA
and rB∩A⊂rA.

Proof.

The first statement follows easily from Lemma 1.9.
The second follows from the first and the relations
rA=¯¯¯¯¯¯¯¯¯¯¯¯¯¯R+FA
and rB=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯R+FB
(Proposition 2.16).
∎

Proposition 2.20.

Let p∈(1,∞).
Let B be a nonunital approximately unital Banach algebra,
and let A⊆B be a
closed subalgebra
which contains a cai for B.
Then:

A1⊆B1 isometrically.

FA=FB∩A
and rA=rB∩A.

Every state or quasistate
on A may be extended to a state or quasistate on B.

Proof.

Part (1) is Lemma 1.10.
That FA=FB∩A
is immediate from (1),
and now rA=rB∩A
by e.g. Proposition 2.16.
Part (3) is obvious from (1),
Definition 2.6, and the Hahn-Banach Theorem.
∎

Lemma 2.21.

Suppose that an Arens regular
Banach algebra A has a cai
and also has a real positive approximate identity.
Then A has a cai in FA.
If in addition A has a countable bounded approximate identity,
then A has a cai in FA
which is a sequence.

Proof.

Corollary 3.9 of [7] implies that
A has an approximate identity in FA.
Since FA is bounded,
one may then use the argument in the second paragraph
of the proof of [5, Proposition 6.13]
to see that A has a cai (et)t∈Λ in FA.
If in addition A has a countable bounded approximate identity,
then one can use Corollary 32.24 of [27]
and its analog on the right (see also Theorem 4.4 in [7]) to find
x,y∈A with A=¯¯¯¯¯¯¯xA=¯¯¯¯¯¯¯Ay.
Choose t1,t2,…∈Λ
with t1<t2<⋯ and
∥ftkx−x∥+∥yftk−y∥<2−k; then
(ftk) is a countable cai in FA.
∎

Corollary 2.22.

Suppose that A is an approximately unital Arens regular
Banach algebra.

If 1A∗∗ is a weak* limit
of
a bounded net of real positive elements in A, then A has a real positive cai.

If A has one of the Kaplansky density type properties
(2)
or (3)
listed in Subsection 4.7,
then A has a real positive cai.

Proof.

For (1),
by a standard convexity argument,
or e.g. [7, Lemma 2.1],
A has a real positive bounded approximate
identity.
It follows from Lemma 2.21
that A has a cai in FA.

Part (2)
follows because these conditions
imply that the hypothesis in (1) holds.
(See also the proof of Proposition 6.4 in [7].)
∎

2.3. More on the multiplier unitization

Lemma 2.23.

Let E be a Banach space.
Suppose that A is
a nonunital closed approximately unital subalgebra of B(E)
which acts nondegenerately on E.
Then the multiplier unitization of
A is isometrically isomorphic to A+C1E,
where 1E is the identity operator on E.

Proof.

For a,c∈A and λ∈C,
we clearly have

∥ac+λc∥=∥(a+λ1E)c∥≤∥a+λ1E∥∥c∥.

So ∥a+λ1∥A1≤∥a+λ1E∥.
The reverse inequality
follows from the fact that if (et)t∈Λ
is a cai for A,
then aet+λet→a+λ1E
in the strong operator topology on B(E).
∎

Lemma 2.24.

Suppose that A is an approximately unital
Arens regular Banach algebra,
and let e=(et)t∈Λ be a cai for A.
Then:

The multiplier unitization of
A is isometrically isomorphic to A+C1A∗∗ in A∗∗.

With Se(A) as defined
in Definition 2.6,
and identifying A∗
with the weak* continuous functionals on A∗∗,
we have

Se(A)={ω∈S(A∗∗):ω is weak* %
continuous}

(the normal state space of A∗∗).

Se(A) and S(A) both span A∗,
and both separate the points of A.

In the notation found before Lemma 2.6
of [7]
and in Definition 2.12,
we have

reA=rAandceA∗=cA∗.

If A is also nonunital
then {φ|A:φ∈S(A1)} is the
weak* closure in A∗ of any one of the following
sets in Definition 2.6:
S(A), Se(A), Q(A), and Qe(A).

Proof.

The proof of (1)
is essentially the same as the proof of Lemma 2.23:
for a,c∈A and λ∈C,
clearly

∥ac+λc∥=∥(a+λ1A∗∗)c∥≤∥a+λ1A∗∗∥∥c∥.

So ∥a+λ1∥A1≤∥a+λ1A∗∗∥.
The reverse inequality
follows from the fact that
if (et)t∈Λ is a cai,
then Lemma 1.12
implies that aet+λet→a+λ1A∗∗ weak*.

For (2),
since et→1 weak* in A∗∗
by Lemma 1.12,
it is clear that weak* continuous states on A∗∗
restrict to elements of Se(A).
For the reverse inclusion,
let ω∈Se(A).
Then ω∗∗ is a weak* continuous functional on A∗∗
and ∥ω∗∗∥=1.
That ω∗∗(1)=1
follows from weak* continuity of ω∗∗
and the weak* convergence et→1.

The assertion about Se(A) in
(3)
follows from part (2)
and Theorem 2.2 of [37],
according to which the normal state space of A∗∗
spans A∗
and separates the points of A.
The second assertion in (3)
follows from the first assertion
and the inclusion Se(A)⊆S(A),
which is in Lemma 2.2 of [7].

We prove (4).
We need only prove
reA⊆rA,
since the reverse inclusion holds by definition,
and equality implies
cA∗=ceA∗
by definition.
So let
a∈reA
and let ω∈S(A).
By definition,
ω extends to a state ω1 on A1.
By part (1)
we have A1⊆A∗∗,
so the Hahn-Banach Theorem provides an extension of ω1
to a state φ on A∗∗.
Use weak* density of the normal states in S(A∗∗)
(which follows from Theorem 2.2 of [37])
to find a net (φt)t∈Λ
in the normal state space of A∗∗
which converges weak* to φ.
Now
Re(ω(a))=limtRe(φt(a))≥0.
So a∈rA.

It follows from [7, Lemma 2.6] that, with overlines
denoting weak* closures, we have

¯¯¯¯¯¯¯¯¯¯¯¯S(A)=¯¯¯¯¯¯¯¯¯¯¯¯¯Q(A)⊆{φ|A:φ∈S(A1)}.

Also, {φ|A:φ∈S(A1)}
is shown to be weak* closed in the proof of that
lemma.

Now suppose that φ∈S(A1)
and set ψ=φ|A.
Use the Hahn-Banach Theorem to extend φ
to a state ρ on A∗∗.
Use again weak* density of the normal states in S(A∗∗)
to find a net (ψt)t∈Λ
in the normal state space of A∗∗
which converges weak* to ρ.
Set φt=ψt|A for t∈Λ.
For a∈A we then have

φt(a)=ψt(a)→ψ(a)=φ(a).

By part (2),
this shows that ψ is in the weak* closure
of Se(A).
Since Se(A)⊆S(A)⊆Q(A)
and Se(A)⊆Qe(A),
the assertion follows.
∎

The set rA∗∗, as
defined on p. 11 of [7],
may be a proper subset
of the accretive elements in A∗∗, even for
approximately unital Lp-operator algebras.
In fact,
the identity e of A∗∗ is certainly accretive in A∗∗,
but need not be accretive in (A1)∗∗.
(Equivalently,
by Lemma 2.27 (4),
we need not have ∥1−e∥≤1.)
This happens for A=K(Lp([0,1])),
by Proposition 3.10.
However it follows from the later result Proposition 4.26
(and Proposition 4.24 (2))
that rA∗∗, as
defined on p. 11 of [7],
equals the accretive elements in A∗∗
if A is a scaled approximately unital Lp-operator algebra.

Remark 2.25.

The sets Se(A) and Qe(A)
are easily seen to be convex in A∗.
We do not know whether S(A) and Q(A) are necessarily convex
if A is a general approximately unital Arens regular Banach algebra,
since convex combinations
of norm 1 functionals may have norm strictly less than 1.
However they are convex if A is
an approximately unital Lp-operator algebra,
since Corollary 4.25 (1) below
implies convexity of S(A), and this implies convexity of Q(A).

Proposition 2.26.

Let p∈(1,∞).
The multiplier unitization of an approximately unital
Lp-operator algebra
is an Lp-operator algebra.

Proof.

This follows from Lemma 2.24 (1) and the fact
(Lemma 2.1 (3))
that
biduals of Lp-operator algebras are Lp-operator algebras
(or from Lemmas 2.23 and 2.31).
∎

Similarly, for p∈(1,∞)
the multiplier unitization of an approximately unital
SQp-operator algebra
is an SQp-operator algebra.

The multiplier algebra M(A),
and the left and right multiplier algebras
LM(A) and RM(A),
of an approximately Lp-operator algebra
may be defined to be subsets of A∗∗
just as in the operator algebra case.
Then the multiplier unitization A1
is contained in M(A)
isometrically and unitally.
If A is represented isometrically and nondegenerately on Lp(X)
then, just as in the operator algebra case,
M(A), LM(A),
and RM(A) may be identified isometrically
as Banach algebras with the usual subalgebras of B(Lp(X)).
See Theorem 3.19 in [25], and the discussion in that paper.
One can also, for example,
copy the proof of Theorem 2.6.2 of [6]
for LM(A),
and later results in Section 2.6 of [6]
for RM(A) and M(A).

In particular,
M(A), LM(A),
and RM(A)
are all unital Lp-operator algebras.
Similarly, LM(A)
can be identified with the
algebra of bounded right A-module endomorphisms of A,
as usual.
One may also check that
the useful principle in [6, Proposition 2.6.12]
holds for approximately Lp-operator algebras,
with the same proof.
(Also see Theorem 3.17 in [25].)

2.4. Idempotents

We recall that if A is a unital Banach algebra,
then an idempotent e∈A
is called bicontractive
if ∥e∥≤1 and ∥1−e∥≤1.
We collect some standard facts related to bicontractive idempotents.
We say that an element s of a unital Banach algebra A
is an invertible isometry
if s is invertible, ∥s∥=1, and ∥s−1∥=1.

Lemma 2.27.

Let A be a unital Banach algebra
and let e∈A be a hermitian idempotent.
Then 1−2e is an invertible isometry of order 2.

Let A be a unital Banach algebra.
Then every hermitian idempotent in A is bicontractive.

Let p∈[1,∞),
let (X,μ) be a measure space,
and let e∈B(Lp(X,μ))
be an idempotent.
Then e is bicontractive if and only if
1−2e is an invertible isometry.

Let A be a unital Banach algebra
and let e∈A be an idempotent.
Then e is real positive if and only if 1−e
is contractive (∥1−e∥≤1).

The converse of (2)
is false,
even in Lp operator algebras.
See Lemma 6.11 of [46],
which is just the idempotent e2 of Example 3.2
for p≠2.

Part (3)
fails in general unital Banach algebras.
This failure is well known,
and our Example 4.7
contains an explicit counterexample.